Interpolated particle hair are spread all over the object
After looking at the code for quite a bit, I can answer my own question. Interpolated doesn't work like this. What it interpolates is not the position where the children are placed. It is used to interpolate the form (length, curving...) of the hairs from the closest neighbors, but the placement is random. Confining that seems only be possible by specifying a vertex group for the density and using Weight Paint on it.